Independence Number and Maximal Chromatic Polynomials of Connected Graphs

Let \({\mathcal {C}}_{k}(n)\) denote the family of all connected graphs of order n with chromatic number k. In this paper we show that the conjecture proposed by Tomescu which if \(x\ge k\ge 4\) and \(G\in {\mathcal {C}}_{k}(n)\), then

$$\begin{aligned} P(G,x)\le (x)_{k} (x-1)^{n-k} \end{aligned}$$

holds under the additional condition that G has an independent cut-set T of size at most 2 such that the number of components in \(G{\setminus } T\) is equal to the independence number of G.